Video: Simplifying and Evaluating Algebraic Fractions Using Laws of Exponents

Simplify (3๐‘ฅ๐‘ฆ)ยณ/(โˆ’๐‘ฅยฒ๐‘ฆ)ยณ, where ๐‘ฅ๐‘ฆ โ‰  0, and evaluate this expression when ๐‘ฅ = โˆ’1 and ๐‘ฆ = โˆ’2.

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Video Transcript

Simplify three ๐‘ฅ๐‘ฆ to the power of three over negative ๐‘ฅ squared ๐‘ฆ to the power of three, where ๐‘ฅ๐‘ฆ is not equal to zero, and evaluate this expression when ๐‘ฅ equals negative one and ๐‘ฆ equals negative two.

Now the first stage of this problem is to actually simplify three ๐‘ฅ๐‘ฆ all cubed over negative ๐‘ฅ squared ๐‘ฆ all cubed. Well, first of all, what weโ€™re gonna do is actually multiply out the parentheses. So weโ€™re gonna start with the numerator. And what we can actually think is if we have three ๐‘ฅ๐‘ฆ all cubed, and that actually means each of these terms cubed, so itโ€™s like three cubed multiplied by ๐‘ฅ cubed multiplied by ๐‘ฆ cubed, which is gonna give us 27๐‘ฅ cubed ๐‘ฆ cubed. Thatโ€™s cause three cubed, three multiplied by three is nine, multiplied by another three is 27.

And then weโ€™ve got ๐‘ฅ cubed ๐‘ฆ cubed. Well, then weโ€™re gonna have the same for the denominator. So weโ€™ve got negative ๐‘ฅ squared all cubed multiplied by ๐‘ฆ cubed. Well, therefore, what weโ€™re gonna use is actually one of our index laws to rewrite this because weโ€™ve got an index law that tells us that if we have ๐‘ฅ to the power of ๐‘Ž then to the power of ๐‘, this is equal to ๐‘ฅ power of ๐‘Ž multiplied by ๐‘.

So therefore, our denominator is gonna be negative ๐‘ฅ to the power of six, because we had ๐‘ฅ power of two to the power of three, multiply three and two, we get six, then ๐‘ฆ cubed. So now to actually simplify this, what weโ€™re gonna do is apply another index law, which tells us that if we have ๐‘ฅ to the power of ๐‘Ž divided by ๐‘ฅ to the power of ๐‘, then this is gonna be equal to ๐‘ฅ to the power of ๐‘Ž minus ๐‘.

So therefore, our fully simplified answer is gonna be negative 27๐‘ฅ power of negative three. Thatโ€™s because if we have ๐‘ฅ to the power of three divided by ๐‘ฅ to the power of six, weโ€™ll actually subtract them from each other. So we have three minus six, which gives us negative three. And then if you have ๐‘ฆ to the power of three divided by ๐‘ฆ to the power of three, we subtract three from three, which gives us zero. So weโ€™re left with negative 27๐‘ฅ to the power negative three.

Okay, so great! Thatโ€™s the first part of the question answered. But now what we need to do is actually evaluate the expression when ๐‘ฅ is equal to negative one and ๐‘ฆ is equal to negative two. So therefore, to actually evaluate this expression and find out how much it is when ๐‘ฅ is equal to negative one and ๐‘ฆ is equal to negative two, all we have to do is actually substitute in ๐‘ฅ is equal to negative one. And this is because when we simplified it, we actually donโ€™t have any ๐‘ฆ terms left.

So therefore, we can actually rewrite this as negative 27 over negative one all cubed. And thatโ€™s cause we substituted in ๐‘ฅ as negative one. And the reason we can write it like this is because weโ€™ve got another relationship we know, which is that ๐‘ฅ to the power of negative ๐‘Ž is equal to one over ๐‘ฅ to the power of ๐‘Ž. So this is gonna give us negative 27 divided by negative one. And thatโ€™s because negative one cubed is negative one. So therefore, as a negative divided by a negative is positive, it just gives us an answer of 27.

So now even though weโ€™ve actually fully simplified and found the answer, what weโ€™re gonna do is actually check because what we can do is check by substituting in ๐‘ฅ equals negative one and ๐‘ฆ equals negative two into the original expression. And we can actually see there if weโ€™ve actually managed to fully simplify correctly and find the correct value, because this should give us the same value.

So therefore, if we substitute in our values for ๐‘ฅ and ๐‘ฆ, weโ€™ll get three multiplied by negative one multiplied by negative two all cubed over negative then negative one squared multiplied by negative two, and then this is all cubed. So therefore, this is gonna give us six cubed. Thatโ€™s because we had three multiplied by negative one, which is negative three, multiplied by negative two, which is gonna give us six, cause a negative multiplied by a negative is positive, so six cubed, over, then weโ€™ve got two cubed. Thatโ€™s because we had negative negative one squared. Well, negative one squared is positive one. But we make that negative cause the negative sign in front, so weโ€™ve got negative one multiplied by negative two, which is positive two. And then this is all cubed, which is gonna give us 216 over eight, which gives us 27, which is the answer we got when we worked out the first method.

So great! What we can say is that, fully simplified, three ๐‘ฅ๐‘ฆ to the power of three over negative ๐‘ฅ squared ๐‘ฆ to the power of three is negative 27๐‘ฅ power of negative three. And the value of this expression when ๐‘ฅ is equal to negative one and ๐‘ฆ is equal to negative two is 27.

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